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Unformatted text preview: Lecture Notes 11 Econ 410 Introduction to Econometrics 1 IV Regression: Part II Comparing the bias of the OLS and TSLS estimators in large samples (Note: in order to keep things simple, the following arguments are based on the linear regression model with one endogenous variable and one instrument, but they can be extended to the general IV regression model.) Consider the following model: i i i u X Y + + = 1 where ( ) , i i u X corr ( i X is an endogenous variable). In this case, the conditional mean assumption doesnt hold, because ( ) | i i X u E . OLS estimator 1 The OLS estimator for this model is biased and inconsistent (see Chapter 6 and Lecture Notes 6). Bias The formula for the OLS estimator can be written as: ( ) ( ) = = + = n i i i n i i X X u X X 1 2 1 1 1 so that: ( ) ( ) ( ) + = = = n i i i n i i X X u X X E E 1 2 1 1 1 Since the conditional mean assumption doesnt hold, the expectation in the right hand side is not equal to zero. Then, the OLS estimator is biased and the bias is equal to: ( ) ( ) = = = n i i i n i i X X u X X E bias 1 2 1 Lecture Notes 11 Econ 410 Introduction to Econometrics 2 Inconsistency From the formula for the OLS estimator, we have that: ( ) ( ) ( ) ( ) X u X p n i i i n i i u X corr u X X X u X X , , cov 1 2 1 1 2 1 1 1 + = + + = = = Since ( ) , i i u X corr , the OLS estimator is inconsistent. In particular, ( ) X u u X corr , is the bias that persists in large samples. In other words, even if the sample is very large, 1 will not be close to 1 and the difference (the bias) between the estimated and the true coefficient will be ( ) X u u X corr , . TSLS estimator TSLS 1 If the instrument i Z is valid, the TSLS estimator for this model is consistent and, in very large samples, it is approximately unbiased. Bias The formula for the TSLS estimator can be written as (see Appendix 12.3 in the book): ( ) ( ) ( ) = = + = n i i i i n i i TSLS X X Z Z u Z Z 1 1 1 1 From this formula we have: ( ) ( ) ( ) ( )...
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